Cooperation is a representative altruistic behavior in which individuals contribute public goods to benefit their neighborhoods and even larger communities in social networks. The defective behavior is more likely to bring higher payoffs than the cooperative behavior, which makes the cooperative behavior hard to maintain and sustain. Many mechanisms were proposed to promote cooperation within a social dilemma, in which some recent studies introduced the impact of dynamically changing environments on players’ payoffs and strategies in social-ecological systems, and evolutionary-ecological systems. However, degree heterogeneity, an important structural property of many real-world complex networks such as social networks, academic collaboration networks, and communication networks, is rarely explored and studied in such eco-evolutionary games. In this research, we propose a Public Goods Game model on social networks with environmental feedback and analyze how the environmental factor and network structure affect the evolution of cooperation. It is found that as the initial environmental factors and the cooperation-enhancement defection-degradation ratio increase, the steady cooperation level of the social network significantly increases, and the dynamic environment will eventually evolve into a high-return environment; On the other hand, even if the initial environmental benefit coefficient is high, when the cooperation-enhancement defection-degradation ratio is less than a threshold, the dynamic environment will gradually degrade into a low-return environment. The steady cooperation level of the social network first gradually increases as the network structure becomes more heterogeneous, but it will decrease once the heterogeneity of the social network exceeds a certain threshold.

1.
E.
Lieberman
,
C.
Hauert
, and
M. A.
Nowak
, “
Evolutionary dynamics on graphs
,”
Nature
433
,
312
316
(
2005
).
2.
M. A.
Nowak
,
Evolutionary Dynamics: Exploring the Equations of Life
(
Harvard University Press
,
Cambridge
,
2006
).
3.
G.
Szabó
and
G.
Fath
, “
Evolutionary games on graphs
,”
Phys. Rep.
446
,
97
216
(
2007
).
4.
H.
Guo
,
C.
Mu
,
Y.
Chen
,
C.
Shen
,
S.
Hu
, and
Z.
Wang
, “Multi-agent, human-agent and beyond: A survey on cooperation in social dilemmas,” arXiv:2402.17270 (2024).
5.
Z.
Wang
,
M.
Jusup
,
R.-W.
Wang
,
L.
Shi
,
Y.
Iwasa
,
Y.
Moreno
, and
J.
Kurths
, “
Onymity promotes cooperation in social dilemma experiments
,”
Sci. Adv.
3
,
e1601444
(
2017
).
6.
Z.
Wang
,
M.
Jusup
,
L.
Shi
,
J.-H.
Lee
,
Y.
Iwasa
, and
S.
Boccaletti
, “
Exploiting a cognitive bias promotes cooperation in social dilemma experiments
,”
Nat. Commun.
9
,
2954
(
2018
).
7.
Y.
Liu
,
C.
Yang
,
K.
Huang
, and
Z.
Wang
, “
Swarm intelligence inspired cooperation promotion and symmetry breaking in interdependent networked game
,”
Chaos
29
,
043101
(
2019
).
8.
A.
Pal
and
S.
Sengupta
, “
Network rewiring promotes cooperation in an aspirational learning model
,”
Chaos
32
,
023109
(
2022
).
9.
J.
Quan
,
X.
Chen
,
W.
Yang
, and
X.
Wang
, “
Cooperation dynamics in spatial public goods games with graded punishment mechanism
,”
Nonlinear Dyn.
111
,
8837
8851
(
2023
).
10.
H.
Ohtsuki
,
C.
Hauert
,
E.
Lieberman
, and
M. A.
Nowak
, “
A simple rule for the evolution of cooperation on graphs and social networks
,”
Nature
441
,
502
505
(
2006
).
11.
C.
Efferson
,
C. P.
Roca
,
S.
Vogt
, and
D.
Helbing
, “
Sustained cooperation by running away from bad behavior
,”
Evol. Hum. Behav.
37
,
1
9
(
2016
).
12.
Y.
Bramoullé
and
R.
Kranton
, “
Public goods in networks
,”
J. Econ. Theory
135
,
478
494
(
2007
).
13.
C.
Deng
,
L.
Wang
,
Z.
Rong
, and
X.
Wang
, “
Cooperation emergence in group population with unequal competitions
,”
Europhys. Lett.
131
,
28001
(
2020
).
14.
S.
Pathak
,
P.
Verma
,
S. K.
Ram
, and
S.
Sengupta
, “
How strategy environment and wealth shape altruistic behaviour: Cooperation rules affecting wealth distribution in dynamic networks
,”
Proc. R. Soc. B: Biol. Sci.
287
,
20202250
(
2020
).
15.
F.
Giardini
,
D.
Vilone
,
A.
Sánchez
, and
A.
Antonioni
, “
Gossip and competitive altruism support cooperation in a public good game
,”
Philos. Trans. R. Soc. B: Biol. Sci.
376
,
20200303
(
2021
).
16.
X.
Wang
and
M.
Perc
, “
Bilateral costly expulsions resolve the public goods dilemma
,”
Proc. R. Soc. A: Math., Phys. Eng. Sci.
477
,
20210627
(
2021
).
17.
S.
Podder
,
S.
Righi
, and
F.
Pancotto
, “
Reputation and punishment sustain cooperation in the optional public goods game
,”
Philos. Trans. R. Soc. B: Biol. Sci.
376
,
20200293
(
2021
).
18.
X.
Wang
,
Z.
Yang
,
G.
Chen
, and
Y.
Liu
, “
Enhancing cooperative evolution in spatial public goods game by particle swarm optimization based on exploration and q-learning
,”
Appl. Math. Comput.
469
,
128534
(
2024
).
19.
A. R.
Tilman
,
J. B.
Plotkin
, and
E.
Akçay
, “
Evolutionary games with environmental feedbacks
,”
Nat. Commun.
11
,
915
(
2020
).
20.
X.
Wang
and
F.
Fu
, “
Eco-evolutionary dynamics with environmental feedback: Cooperation in a changing world
,”
Europhys. Lett.
132
,
10001
(
2020
).
21.
X.
Ma
,
J.
Quan
, and
X.
Wang
, “
Evolution of cooperation with nonlinear environment feedback in repeated public goods game
,”
Appl. Math. Comput.
452
,
128056
(
2023
).
22.
C.
Di
,
Q.
Zhou
,
J.
Shen
,
J.
Wang
,
R.
Zhou
, and
T.
Wang
, “
The coupling effect between the environment and strategies drives the emergence of group cooperation
,”
Chaos, Solitons Fractals
176
,
114138
(
2023
).
23.
Y.
Guo
,
L.
Zhang
,
H.
Li
,
Q.
Dai
, and
J.
Yang
, “
Network adaption based on environment feedback promotes cooperation in co-evolutionary games
,”
Phys. A: Stat. Mech. Appl.
617
,
128689
(
2023
).
24.
W.-J. P.
Chiou
,
A. C.
Lee
, and
C.-F.
Lee
, “
Stock return, risk, and legal environment around the world
,”
Int. Rev. Econ. Finance
19
,
95
105
(
2010
).
25.
R.
Pindyck
and
D.
Rubinfeld
,
Microeconomics
(Pearson Education, Londorn, 2018).
26.
J. S.
Weitz
,
C.
Eksin
,
K.
Paarporn
,
S. P.
Brown
, and
W. C.
Ratcliff
, “
An oscillating tragedy of the commons in replicator dynamics with game-environment feedback
,”
Proc. Natl. Acad. Sci. U.S.A.
113
,
E7518
E7525
(
2016
).
27.
I.
Farahbakhsh
,
C. T.
Bauch
, and
M.
Anand
, “
Modelling coupled human–environment complexity for the future of the biosphere: Strengths, gaps and promising directions
,”
Philos. Trans. R. Soc. B: Biol. Sci.
377
,
20210382
(
2022
).
28.
Z.
Wang
,
Z.
Song
,
C.
Shen
, and
S.
Hu
, “Emergence of punishment in social dilemma with environmental feedback,” in Proceedings of the 37th AAAI Conference on Artificial Intelligence (AAAI, 2023), Vol. 37, pp. 11708–11716.
29.
Y.
Jiang
,
X.
Wang
,
L.
Liu
,
M.
Wei
,
J.
Zhao
,
Z.
Zheng
, and
S.
Tang
, “
Nonlinear eco-evolutionary games with global environmental fluctuations and local environmental feedbacks
,”
PLoS Comput. Biol.
19
,
e1011269
(
2023
).
30.
Ö.
Bodin
, “
Collaborative environmental governance: Achieving collective action in social-ecological systems
,”
Science
357
,
eaan1114
(
2017
).
31.
Z.
Wang
,
M.
Jusup
,
H.
Guo
,
L.
Shi
,
S.
Geček
,
M.
Anand
,
M.
Perc
,
C. T.
Bauch
,
J.
Kurths
,
S.
Boccaletti
, and
H. J.
Schellnhuber
, “
Communicating sentiment and outlook reverses inaction against collective risks
,”
Proc. Natl. Acad. Sci. U.S.A.
117
,
17650
17655
(
2020
).
32.
E.
Ostrom
, “
A general framework for analyzing sustainability of social-ecological systems
,”
Science
325
,
419
422
(
2009
).
33.
S.
Partelow
, “
A review of the social-ecological systems framework
,”
Ecol. Soc.
23
(4),
36
(
2018
).
34.
D. J. D. S.
Price
, “
Networks of scientific papers: The pattern of bibliographic references indicates the nature of the scientific research front
,”
Science
149
,
510
515
(
1965
).
35.
D. D. S.
Price
, “
A general theory of bibliometric and other cumulative advantage processes
,”
J. Am. Soc. Inf. Sci.
27
,
292
306
(
1976
).
36.
A.-L.
Barabási
and
R.
Albert
, “
Emergence of scaling in random networks
,”
Science
286
,
509
512
(
1999
).
37.
M. E.
Newman
, “
Assortative mixing in networks
,”
Phys. Rev. Lett.
89
,
208701
(
2002
).
38.
M. E.
Newman
, “
Mixing patterns in networks
,”
Phys. Rev. E
67
,
026126
(
2003
).
39.
Z.
Rong
,
X.
Li
, and
X.
Wang
, “
Roles of mixing patterns in cooperation on a scale-free networked game
,”
Phys. Rev. E
76
,
027101
(
2007
).
40.
M. W.
Macy
and
A.
Flache
, “
Learning dynamics in social dilemmas
,”
Proc. Natl. Acad. Sci. U.S.A.
99
,
7229
7236
(
2002
).
41.
S. S.
Izquierdo
,
L. R.
Izquierdo
, and
N. M.
Gotts
, “
Reinforcement learning dynamics in social dilemmas
,”
J. Artifi. Soc. Soc. Simul.
11
,
1
(
2008
).
42.
Z.
Wang
,
C.
Mu
,
S.
Hu
,
C.
Chu
, and
X.
Li
, “Modelling the dynamics of regret minimization in large agent populations: A master equation approach,” in Proceedings of the 31st International Joint Conference on Artificial Intelligence (IJCAI, 2022), pp. 534–540.
43.
R. R.
Bush
and
F.
Mosteller
, “
A stochastic model with applications to learning
,”
Ann. Math. Stat.
24
,
559
585
(
1953
).
44.
P.
Erdős
A.
Rényi
et al., “
On the evolution of random graphs
,”
Publ. Math. Inst. Hungar. Acad. Sci.
5
,
17
60
(
1960
).
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